OFFSET
1,2
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The prime signature of a number is the multiset of multiplicities (or exponents) in its prime factorization.
Also Heinz numbers of integer partitions where the multiplicity of k is at least k (A117144). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Closed under multiplication.
Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^(k-1) * (prime(k)-1)) = 2.35782843100111139159... - Amiram Eldar, Nov 23 2020
EXAMPLE
The sequence of terms together with their prime indices begins as follows. For example, 36 = prime(1) * prime(1) * prime(2) * prime(2) is a term because the prime multiplicities are {2,2}, which are greater than or equal to the prime indices {1,2}.
1: {}
2: {1}
4: {1,1}
8: {1,1,1}
9: {2,2}
16: {1,1,1,1}
18: {1,2,2}
27: {2,2,2}
32: {1,1,1,1,1}
36: {1,1,2,2}
54: {1,2,2,2}
64: {1,1,1,1,1,1}
72: {1,1,1,2,2}
81: {2,2,2,2}
108: {1,1,2,2,2}
125: {3,3,3}
128: {1,1,1,1,1,1,1}
MAPLE
q:= n-> andmap(i-> i[2]>=numtheory[pi](i[1]), ifactors(n)[2]):
select(q, [$1..3000])[]; # Alois P. Heinz, Mar 08 2019
MATHEMATICA
Select[Range[1000], And@@Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>k>=PrimePi[p]]&]
seq[max_] := Module[{ps = {2}, p, s = {1}, s1, s2, emax}, While[ps[[-1]]^Length[ps] < max, AppendTo[ps, NextPrime[ps[[-1]]]]]; Do[p = ps[[k]]; emax = Floor[Log[p, max]]; s1 = Join[{1}, p^Range[k, emax]]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &]; s = Union[s, s2], {k, 1, Length[ps]}]; s]; seq[3000] (* Amiram Eldar, Nov 23 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 08 2019
STATUS
approved